## Mathematical Methods (as MAGIC022)

 Unit code: MATH64051 Credit Rating: 15 Unit level: Level 6 Teaching period(s): Semester 1 Offered by School of Mathematics Available as a free choice unit?: N

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#### Aims

To provide training in a variety of mathematical methods and techniques in order that they may be applied to a wide range of problems in applied mathematics and numerical analysis.

#### Overview

This unit will teach a variety of powerful mathematical methods that can be used in order to solve mathematical problems that arise in numerous applications. This will include both exact and approximate solutions. In the case of the latter, asymptotic and perturbation methods will be developed to solve partial differential equations and determine approximations for integrals.

#### Learning outcomes

On successful completion of this course unit students will be able to:

• Apply non-dimensionalisation techniques and interpret the importance of non-dimensional parameters.
• Identify invariance properties of operators and explore similarity type solutions.
• Recall basic complex variable theory and be able to compute residues.
• Use results from complex variable theory in order to evaluate integrals.
• Discuss properties of certain special functions and be able to manipulate them.
• Identify properties of asymptotic expansions and be able to use them effectively.
• Recall the properties of Geometric progressions, Taylor series, and use them to derive asymptotic expansions of integrals.
• Discuss the properties of various integral transforms and use them to solve problems.
• Describe the properties of the Mellin transform use it to estimate integrals.
• Use asymptotic and perturbation methods in order to solve ordinary and partial differential equations approximately.

#### Assessment methods

• Other - 20%
• Written exam - 80%

#### Assessment Further Information

• Mid-semester coursework as a take-home test: 20%
• End of semester examination: three hours weighting 80%

#### Syllabus

Topic 1-9 below are the same as the MAGIC022 course with 10-14 being the additional material on applications.

• Non-dimensionalization and scaling.[1]
• Advanced differential equations. Series solution, classification of singularities. Properties near ordinary and regular singular points. Approximate behaviour near irregular singular points. Method of dominant balance. Airy, Gamma and Bessel functions. [3]
• Asymptotic methods. Boundary layer theory. Regular and singular perturbation problems. Uniform approximations. Interior layers. LG approximation, WKBJ method. Multiple scales. [3]
• Generalized functions. Basic definitions and properties. [2].
• Complex analysis. Revision of background material. Laurent expansions. Singularities, Cauchy?s theorem. Residue calculus. Branch points and cuts. Plemelj formulae. [3]
• Transform methods. Fourier transform, FT of generalized functions. Laplace transform. Properties of Gamma function. Mellin transform. Analytic continuation of Mellin transforms. .[3]
• Similarity solutions of pdes [1]
• Asymptotic expansion of integrals Laplace?s method. Watson?s Lemma. Methods of stationary phase and steepest descent. Estimation using Mellin transform technique.[2]
• Conformal mappings [2]
• The MSc course will then showcase these methods with applications to 2-3 of the following areas:[10]

- Greens functions for ODEs

- Thin aerofoil theory

- Application of steepest descent to a classical problem, e.g. water waves

- Application of multiple scales to homogenization

- Transforms to solve BVPs and use of complex analysis to invert

- Use of WKB for e.g. geometric optics, caustics, etc

#### Recommended reading

• R. Wong, Asymptotic Approximation of Integrals, Academic Press 1989.
• E.J. Hinch, Perturbation Methods, Cambridge 1991.
• O.M. Bender and S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers.
• M. Van Dyke, Perturbation Methods in Fluid Mechanics, Academic Press 1964.
• J. Kevorkian and J.D. Cole, Perturbation Methods in Applied Mathematics, Springer 1985.
• A.H. Nayfeh, Perturbation Methods, Wiley 1973

#### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

#### Study hours

• Lectures - 30 hours
• Tutorials - 6 hours
• Independent study hours - 114 hours

#### Teaching staff

Jitesh Gajjar - Unit coordinator